@mrxtraf
In the private key group (mod n) we can add, negate, and invert - this allows for multiplication and division.
In the public key group (elliptic curve mod p of size n) we can add, negate, and double only. This leads to multiplication by a scalar.
One public key corresponds to exactly one private key, and vice versa. The proof is very easy. Let G is the generator of secp256k1. Let P=k*G is a point on the curve. Let also P=k'*G. Then (k-k')*G=O => (k-k') divides n. But n is prime, hence k=k' (mod n).
That is, you can’t divide the public key by 10?
Give me any public key from which you know the private key, I will divide it by 10. And I will give in return the result in the form of a public key. And you yourself divide the private key by 10, get the public key from it and compare.